Can the shape of curved space time under influence of mass be closely modelled by any function?

Like, without getting into tensors and Euclidean/non-euclidean geometry, can I make a function (in one dimension), something of the form $y=ax^n+...$ to closely tell the curvature of space time?

  • $\begingroup$ Other than something like $R^{abcd}R_{abcd}$? $\endgroup$ – Jerry Schirmer Jun 23 at 19:21
  • $\begingroup$ The Kretschmann scalar for a Schwarzschild metric has a simple $1/r^6$ dependence. $\endgroup$ – G. Smith Jun 23 at 19:32

The closest thing I'm aware of is Flamm's paraboloid, which is an embedding of the curvature of the spatial part of the Schwarzschild metric in an ambient 3D Euclidean space.

However, it should be noted that this only accounts for a 2D spatial slice of Schwarzschild; in effect, it only tells you about the curvature of space, not the curvature of spacetime. In particular, the geodesics on Flamm's paraboloid (the paths followed by freefalling objects) are not the paths that would be followed by real objects in Schwarzschild spacetime, since real objects are moving in time as well as in space.

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